Painlev\'e analysis of the Bryant Soliton

TL;DR

This paper performs a Painlevé analysis on differential equations related to steady and expanding rotationally symmetric gradient Ricci solitons on Euclidean space, highlighting special dimensions with unique properties.

Contribution

It identifies specific dimensions where the differential systems exhibit special integrability properties, notably in steady and expanding Ricci solitons.

Findings

Dimension 2 is uniquely distinguished for the expanding soliton.

Dimensions of the form n=k^2+1 are special for the steady case.

Certain dimensions (2, 5, 10) are particularly distinguished in the steady case.

Abstract

We carry out a Painlev\'e analysis of the systems of differential equations corresponding to the steady and the expanding, rotationally symmetric, gradient Ricci solitons on $\mathbb{R}^n$. For the steady case, dimensions of the form $n=k^2+1$ are singled out, with dimensions 2, 5, and 10 being particularly distinguished. Only dimension 2 is singled out for the expanding soliton.